Remark 5.12 We remark finally that there is a very substantial class of symmetric spaces \(E\subseteq S(\tau )\)  whose norm is monotone with respect to submajorization. Indeed, if \(E\)  is fully symmetric in the sense that, if \(x\in E, y\in S(\tau )\)  satisfy \(y\prec \prec x\)  then \(y\in E\)  and \(\Vert y\Vert _E \le \Vert x\Vert _E\) , then it is easily seen via remark REF  that
the norm on \(E\)  is monotone with respect to logarithmic submajorization. Such spaces are precisely the exact interpolation spaces for the pair \((L_1(\tau ),{\mathcal {M}})\) . See [1]}, [2]}. If \(E\subseteq S(m)\)  is a fully symmetric space on \([0,\infty )\) , then it is follows directly that \(E(\tau )\subset S(\tau )\)  is fully symmetric. Important special cases occur if \(E\subseteq S(m)\)  is one of the familiar \(L_p\) -spaces, Orlicz spaces or Lorentz spaces. In the case that \({\mathcal {M}}=B({\mathcal {H}})\) , then all trace ideals in the sense of Gohberg and Krein [3]} are fully symmetric. While it is well-known [4]} that there are symmetric spaces on the \([0,\infty )\)  which are not closed subspaces of any fully symmetric space, we do not have an explicit example of a symmetric space on \([0,\infty )\)  whose norm is monotone with respect to logarithmic submajorization
but which is not fully symmetric. This seems to be a difficult problem.

